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I want to understand how to prove results regarding the relationship between the standard and nonstandard normal distributions. In other words, I want to prove the results regarding how to use z tables with nonstandard normal distributions, and finding its p'th percentiles.

What books or articles should I get through to do that? I have taken courses on calculus, real analysis and complex analysis. Do I need anything else to prove the results?

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We have to show if $X\sim \mathcal N(\mu, \sigma^2)$ and $Z=\frac{X-\mu}{\sigma}\sim \mathcal N(0,1)$ then $P(X\leq w)=P(Z\leq \frac{w-\mu}{\sigma})$.

$Z=\frac{X-\mu}{\sigma}\Rightarrow Z\cdot \sigma+\mu=X$

$P(X\leq w)=P(Z\cdot \sigma+\mu\leq w)=P(Z\cdot \sigma\leq w-\mu)=P(Z\leq \frac{w-\mu}{\sigma})$.

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